Cells need to constantly change their change to perform vital functions, such as growth, division, and movement. Cell shape changes are driven by an interplay between the plasma membrane and the internal cytoskeleton, a network of rigid polymers. Our research aim is to elucidate the soft matter physics underlying cell shape changes and cytoskeletal organization. Our approach is to reconstitute simple model systems from a minimal set of purified components. We study aspects such as liquid crystalline ordering of actin filaments, confinement-induced patterning of filaments, and polymer/membrane mechanics. In this talk I will mainly highlight recent findings concerning two active (nonequilibrium) processes that contribute to cytoskeletal organization. First, I will discuss active contractility of actin networks driven by myosin motors. Using fluorescence microscopy, we studied how network connectivity controls the ability of myosin motors to cause large-scale contraction. By systematically tuning the network crosslink density (using fascin), we were able to show that motors contract actin networks only above a sharp threshold in crosslink density. We discovered that right at this threshold, the motors rupture the network into clusters that exhibit a broad distribution of sizes, as expected near a percolation threshold. Surprisingly, we find critical behavior over a broad range of connectivities. We explain this finding with a simple model that accounts for motor-induced crosslink unbinding. Second, I will discuss how actin networks can control the organization of dynamically growing microtubules via a crosslink protein that links growing microtubule plus tips to actin. Aligned arrays of actin bundles, for instance, promote microtubule alignment and stabilize microtubules, whereas isotropic actin networks act as a steric barrier and promote microtubule shrinkage. This crosstalk provides a robust physical mechanism to spatially organize the cytoskeleton in cells.